2015/02/15 by Hédy Attouch, Attouch, Hedy, M. Marques Alves +3
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Optimization and Variational Analysis #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.1502.04286
openalex publication_date 2015/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a Hilbert setting, we introduce a new dynamical system and associated\nalgorithms for solving monotone inclusions by rapid methods.\n Given a maximal monotone operator A, the evolution is governed by the time\ndependent operator I -(I + \λ(t) A)-1, where the positive control\nparameter \λ(t) tends to infinity as t \→ + \∞. The tuning of \n\λ (\⋅) is done in a closed-loop way, by resolution of the algebraic\nequation \λ norm(I + \λ A)-1x -x=\θ, where \θ is\na positive given constant. The existence and uniqueness of a strong global\nsolution for the Cauchy problem follows from Cauchy-Lipschitz theorem. We prove\nthe weak convergence of the trajectories to equilibria, and superlinear\nconvergence under an error bound condition. When A =\∂ f is the\nsubdifferential of a closed convex function f, we show a bigo(1/t2)\nconvergence property of f(x(t)) to the infimal value of the problem. Then, we\nintroduce proximal-like algorithms which can be obtained by time discretization\nof the continuous dynamic, and which share the same fast convergence\nproperties. As distinctive features, we allow a relative error tolerance for\nthe solution of the proximal subproblem similar to the ones proposed in\n~ citeSo-Sv1, So-Sv2, and a large step condition, as proposed\nin~ citeMS1,MS2. For general convex minimization problems, the complexity is\n bigo(1/n2). In the regular case, we show the global quadratic convergence\nof an associated proximal-Newton method.\n